The radiation guiding of a plasma wave wiggler free-electron laser (FEL) in the Compton regime was examined. It was found that a Langmuir wave supported by a plasma cylinder acts as a wiggler for the generation of high-frequency coherent radiation when an annular relativistic electron beam passes through it. The radiation mode in the Compton regime tends to be localized close to the radius of the beam. A normal-mode analysis of this process revealed that the growth rate of the instability increases as the square root of the beam current. The treatment presented is restricted to the case where the radial width of the FEL radiation mode is larger than the beam radius, but smaller than the waveguide radius